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In this paper, the ensembles of repeat multiple- accumulate codes (RAm), which are obtained by interconnecting a repeater with a cascade of m accumulate codes through uniform random interleavers, are analyzed. It is proved that the average spectral shapes of these code ensembles are equal to 0 below a threshold distance epsivm and, moreover, they form a nonincreasing sequence in m converging uniformly to the maximum between the average spectral shape of the linear random ensemble and 0. Consequently the sequence epsivm converges to the Gilbert-Varshamov (GV) distance. A further analysis allows to conclude that if m ges 2 the RAm are asymptotically good and that epsivm is the typical normalized minimum distance when the interleaver length goes to infinity. Combining the two results it is possible to conclude that the typical distance of the ensembles RAm converges to the Gilbert-Varshamov bound.